How do you solve #6x^2 - 7x + 2 = 0# using the quadratic formula?

Question

Eli Assouline · Accepted Answer

The two possible solutions are
#x = 0.667 #
#x = 0.50#

I'll provide the quadratic formula so you can see what I'm doing as I step you through the process:

I think it's worthwhile to mention that #a# is the number that has the #x^2# term associated with it. Thus, it would be #6x^(2)# for this question.#b# is the number that has the #x# variable associated with it and it would be #-7x#, and #c# is a number by itself and in this case it is 2.

Now, we just plug our values into the equation like this:

#x = (- (-7) +- sqrt((-7)^(2) - 4(6)(2)))/(2(6))#

#x = (7 +-sqrt(49-48))/12#

#x = (7 +-1)/12#

For these type of problems, you will obtain two solutions because of the #+-# part. So what you want to do is add 7 and 1 together and divide that by 12:

#x = (7+1)/12#
#x = 8/12 = 0.667#

Now, we subtract 1 from 7 and divide by 12:

#x = (7-1)/12#
# x = 6/12 = 0.50#

Next, plug each value of x into the equation separately to see if your values give you 0. This will let you know if you performed the calculations correctly or not

Let's try the first value of #x# and see if we obtain 0:

#6(0.667)^(2)-7(0.667)+2 = 0#

#2.667 - 4.667 + 2 =0#

#0= 0#

This value of x is correct since we got 0!

Now, let's see if the second value of #x# is correct:

#6(0.50)^(2)-7(0.50)+2 = 0#

1.5 -3.5 +2 = 0#

#0= 0#
That value of x is correct as well!

Thus, the two possible solutions are:

#x = 0.667 #
#x = 0.50#

Julia Adams · Answer

undefined To solve the quadratic equation $6x^2 - 7x + 2 = 0$ using the quadratic formula, which is $x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$, follow these steps:

1. Identify the values of $a$, $b$, and $c$ from the given quadratic equation.
2. Substitute these values into the quadratic formula.
3. Calculate the discriminant, $b^2 - 4ac$.
4. Determine whether the discriminant is positive, negative, or zero.
5. Use the values of $a$, $b$, and $c$ along with the discriminant to find the solutions for $x$.

For the equation $6x^2 - 7x + 2 = 0$, $a = 6$, $b = -7$, and $c = 2$. Substituting these values into the quadratic formula:

$$x = \frac{{-(-7) \pm \sqrt{{(-7)^2 - 4 \cdot 6 \cdot 2}}}}{{2 \cdot 6}}$$

Simplify under the square root:

$$x = \frac{{7 \pm \sqrt{{49 - 48}}}}{{12}}$$
$$x = \frac{{7 \pm \sqrt{1}}}{{12}}$$
$$x = \frac{{7 \pm 1}}{{12}}$$

Now, there are two possible solutions:

$$x_1 = \frac{{7 + 1}}{{12}} = \frac{{8}}{{12}} = \frac{2}{3}$$

$$x_2 = \frac{{7 - 1}}{{12}} = \frac{{6}}{{12}} = \frac{1}{2}$$

Thus, the solutions to the quadratic equation $6x^2 - 7x + 2 = 0$ are $x = \frac{2}{3}$ and $x = \frac{1}{2}$.